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Section 7.3 is devoted to deriving mathematical PDE models for diffusion.
Three different physical applications of increasing complexity are presented in
Sects. 7.3.1 - 7.3.3 . These models, in complete form, are summarized in Sect. 7.3.4 .
Section 7.3.5 discusses scaling, which is a useful tool for simplifying PDE models.
Simple (explicit) numerical methods for one-dimensional diffusion problems are
treated in Sect. 7.4 . Basic finite difference methodology is introduced and applied to
a model problem in Sects. 7.4.1 - 7.4.3 . How to verify that a computer implementa-
tion of the algorithms works is the subject of Sect. 7.4.4 . Some numerical extensions
to handle heterogeneous media are covered in Sect. 7.4.7 .
The methods introduced in Sect. 7.4 are only useful if the time step is below a
certain critical limit. Unconditionally stable methods, without any restriction on the
time step, are introduced in Sect. 7.5 . We first deal with the backward Euler scheme,
then we make extensions to the Crank-Nicolson scheme, and finally we summarize
all our schemes for diffusion problems in terms of the so-called scheme.
As will be evident, simulation of diffusion processes is a wide scientific topic,
including physics, mathematics, numerics, and programming. The mathematical
problem and the basic solution tools are described in Sects. 7.2 - 7.5 . The remaining
chapters aim at providing some understanding of how diffusion models are formu-
lated and some intuition of what kinds of solutions are expected when simulating
such phenomena. All these aspects are crucial for serious simulations of diffusion
problems on a computer.
The information on modeling diffusion processes in Sects. 7.1 and 7.3 is not
required to understand how to simulate diffusion. It is therefore possible to jump
directly to Sect. 7.2 , where the mathematical model of diffusion is listed and then
continue reading about the numerics in Sect. 7.4 , and perhaps continue with Sect. 7.5
if one is interested in more advanced numerical methods for diffusion problems.
7.1
Basics of Diffusion Processes
In this section we give some examples of diffusion phenomena, and we justify why
we later work almost exclusively with one-dimensional models.
7.1.1
Heat Conduction
Suppose you have two pieces of metal. The two pieces have different constant tem-
peratures, so that we can refer to one as hot and the other as cold. Then we bring
the pieces in contact with each other. At the time of contact, the temperature in the
combined piece is discontinuous, since it makes a jump when going from one piece
into the other. Heat will then flow from the hot piece to the cold one. The amount
of heat flow per unit time, the “velocity of heat”, depends on the temperature differ-
ence between the two pieces. The initial jump in the temperature is smoothed, and
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